We report an implementation of the McMurchie-Davidson (MD) algorithm for 3-center and 4-center 2-particle integrals over Gaussian atomic orbitals (AOs) with low and high angular momenta l and varying degrees of contraction for graphical processing units (GPUs). This work builds upon our recent implementation of a matrix form of the MD algorithm that is efficient for GPU evaluation of 4-center 2-particle integrals over Gaussian AOs of high angular momenta (l ≥ 4) [A. Asadchev and E. F. Valeev, J. Phys. Chem. A 127, 10889-10895 (2023)]. The use of unconventional data layouts and three variants of the MD algorithm allow for the evaluation of integrals with double precision and sustained performance between 25% and 70% of the theoretical hardware peak. Performance assessment includes integrals over AOs with l ≤ 6 (a higher l is supported). Preliminary implementation of the Hartree-Fock exchange operator is presented and assessed for computations with up to a quadruple-zeta basis and more than 20 000 AOs. The corresponding C++ code is part of the experimental open-source LibintX library available at https://github.com/ValeevGroup/libintx.
We report an implementation of the McMurchie-Davidson (MD) algorithm for 3-center and 4-center 2-particle integrals over Gaussian atomic orbitals (AOs) with low and high angular momenta l and varying degrees of contraction for graphical processing units (GPUs). This work builds upon our recent implementation of a matrix form of the MD algorithm that is efficient for GPU evaluation of 4-center 2-particle integrals over Gaussian AOs of high angular momenta (l ≥ 4) [A. Asadchev and E. F. Valeev, J. Phys. Chem. A 127, 10889-10895 (2023)]. The use of unconventional data layouts and three variants of the MD algorithm allow for the evaluation of integrals with double precision and sustained performance between 25% and 70% of the theoretical hardware peak. Performance assessment includes integrals over AOs with l ≤ 6 (a higher l is supported). Preliminary implementation of the Hartree-Fock exchange operator is presented and assessed for computations with up to a quadruple-zeta basis and more than 20 000 AOs. The corresponding C++ code is part of the experimental open-source LibintX library available at https://github.com/ValeevGroup/libintx.
High Resolution Image Download MS PowerPoint Slide We report an implementation of the McMurchie–Davidson evaluation scheme for 1- and 2-particle Gaussian AO integrals designed for processors with Single Instruction Multiple Data (SIMD) instruction sets. Like in our recent MD implementation for graphical processing units (GPUs) [ Asadchev, A.; Valeev, E. F. . J. Chem. Phys. 2024, 160, 244109 .], variable-sized batches of shellsets of integrals are evaluated at a time. By optimizing for the floating point instruction throughput rather than minimizing the number of operations, this approach achieves up to 50% of the theoretical hardware peak FP64 performance for many common SIMD-equipped platforms (AVX2, AVX512, NEON), which translates to speedups of up to 30 over the state-of-the-art one-shellset-at-a-time implementation of Obara–Saika-type schemes in Libint for a variety of primitive and contracted integrals. As with our previous work, we rely on the standard C++ programming language─such as the std::simd standard library feature to be included in the 2026 ISO C++ standard─without any explicit code generation to keep the code base small and portable. The implementation is part of the open source LibintX library freely available at https://github.com/ValeevGroup/libintx .
Quantum chemistry offers the formal machinery to derive molecular and physical properties arising from (sub)atomic interactions. However, as molecules of practical interest are largely polyatomic, contemporary approximation schemes such as the Hartree–Fock scheme are computationally expensive due to the large number of electron repulsion integrals (ERIs). Central to the Hartree–Fock method is the efficient computation of ERIs over Gaussian functions (GTO-ERIs). Here, the well-known McMurchie–Davidson method (MD) offers an elegant formalism by incrementally extending Hermite Gaussian functions and auxiliary tabulated functions. Although the MD method offers a high degree of versatility to acceleration schemes through Graphics Processing Units (GPUs), the current GPU implementations limit the practical use of supported values of the azimuthal quantum number. In this paper, we propose a generalized framework capable of computing GTO-ERIs for arbitrary azimuthal quantum numbers, provided that the intermediate terms of the MD method can be stored. Our approach benefits from extending the MD recurrence relations through shells, batches, and triple-buffering of the shared memory, and ordering similar ERIs, thus enabling the effective parallelization and use of GPU resources. Furthermore, our approach proposes four GPU implementation schemes considering the suitable mappings between Gaussian basis and CUDA blocks and threads. Our computational experiments involving the GTO-ERI computations of molecules of interest on an NVIDIA A100 Tensor Core GPU (NVIDIA, Santa Clara, CA, USA) have revealed the merits of the proposed acceleration schemes in terms of computation time, including up to a 72× improvement over our previous GPU implementation and up to a 4500× speedup compared to a naive CPU implementation, highlighting the effectiveness of our method in accelerating ERI computations for both monatomic and polyatomic molecules. Our work has the potential to explore new parallelization schemes of distinct and complex computation paths involved in ERI computation.
In this article, we present an effective approach to calculate quantum chemical two-electron integrals over basis sets consisting of Gaussian-type basis functions on graphical processing unit (GPU). Our framework generates several different variants called routes to the same integral problem with different integral algorithms (McMurchie-Davidson, Head-Gordon-Pople, and Rys) and precision. Each route is benchmarked on more GPU architectures, and with this data, a model is fitted to select the best available route for an integral task given a GPU architecture. Moreover, this approach supports the computation of high angular momentum orbitals up to g effectively on GPU, tested up to cc-pVQZ-sized basis sets. Rigorous analysis is shown regarding the effectiveness of our method. Molecule simulations with several basis sets are measured using NVIDIA GTX 1080 Ti, NVIDIA P100, and NVIDIA V100 cards.
Modern videogames place increasing demands on the computational and graphical hardware, leading to novel architectures that have great potential in the context of high performance computing and molecular simulation. We demonstrate that Graphical Processing Units (GPUs) can be used very efficiently to calculate two-electron repulsion integrals over Gaussian basis functions [Formula: see text] the first step in most quantum chemistry calculations. A benchmark test performed for the evaluation of approximately 10(6) (ss|ss) integrals over contracted s-orbitals showed that a naïve algorithm implemented on the GPU achieves up to 130-fold speedup over a traditional CPU implementation on an AMD Opteron. Subsequent calculations of the Coulomb operator for a 256-atom DNA strand show that the GPU advantage is maintained for basis sets including higher angular momentum functions.
We present a GPU-accelerated software package for the evaluation of analytic two-electron energy and gradient contributions in Hartree-Fock (HF) and Density Functional theory (DFT) calculations. The implementation is provided as a Python library with a C++ backend, enabling straightforward integration into modern computational chemistry and drug-discovery workflows. The code supports single-point energy and nuclear gradient evaluations on both single- and multi-GPU systems, and employs MPI-based parallelization with dynamic load balancing in multi-node environments. We report comprehensive benchmarks demonstrating favorable scaling with respect to system size, as well as high throughput for batched evaluations relevant to molecular dynamics, geometry optimization, and large-scale virtual screening. Parallel execution of a single system was carried out on up to 24 A100 GPUs. The implementation builds on optimized GPU-enabled variants of the LibintX and GauXC libraries to efficiently compute density-fitted Coulomb, semi-numerical exchange, and exchange--correlation contributions.
We present a GPU-accelerated software package for the evaluation of analytic two-electron energy and gradient contributions in Hartree-Fock (HF) and Density Functional theory (DFT) calculations. The implementation is provided as a Python library with a C++ backend, enabling straightforward integration into modern computational chemistry and drug-discovery workflows. The code supports single-point energy and nuclear gradient evaluations on both single- and multi-GPU systems, and employs MPI-based parallelization with dynamic load balancing in multi-node environments. We report comprehensive benchmarks demonstrating favorable scaling with respect to system size, as well as high throughput for batched evaluations relevant to molecular dynamics, geometry optimization, and large-scale virtual screening. Parallel execution of a single system was carried out on up to 24 A100 GPUs. The implementation builds on optimized GPU-enabled variants of the LibintX and GauXC libraries to efficiently compute density-fitted Coulomb, semi-numerical exchange, and exchange--correlation contributions.